# 5.3: Potential Energy Curves

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Harmonic Oscillator

Within the Born-Oppenheimer approximation, each electronic state in a diatomic molecule has a characteristic nuclear potential surface V(R) describing the interatomic potential energy as a function of interatomic distance R. For a bound diatomic molecule with low vibrational energy this interatomic potential V(R) is similar to that of a harmonic oscillator

\[ V\left ( R \right )=\frac{1}{2}k\left ( R-R_e \right )^2 \]

where R-R_{e} is the deviation of the interatomic separation from the equilibrium value R_{e}, and k is a force constant describing the strength of the "harmonic spring" that binds the diatomic molecule together.

The potential energy curve V(R) for a harmonic oscillator is a parabola, and the energy levels are equally spaced, as shown in Figure 5.3.1. The vibrational energy \( E_{\upsilon} \) increases linearly with the vibrational quantum number \( \upsilon \) according to the equation

\[ E_\upsilon=h\nu\left ( \upsilon+\frac{1}{2} \right ) \]

where h is Planck's constant and \(\nu\) is the vibrational frequency. The vibrational frequency \(\nu\) is related to the force constant k and the reduced mass \(\mu\) through the relation

\[ \nu=\frac{1}{2\pi }\sqrt{\left ( \frac{k}{\mu} \right )} \]

where \( \large \mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}} \) for a diatomic molecule where \(m_1\) and \(m_2\) are the atomic masses of the atoms.

** However, the chemical bonds in real diatomic molecules are not harmonic oscillators**.

## Anharmonic Oscillator

As a chemical bond stretches it gets weaker, ultimately resulting in dissociation as R approaches \(\infty \). Also, as the bond shortens, interatomic repulsions raise the potential more rapidly than in the harmonic potential. Thus, __the potential V(R) for a real diatomic molecule is anharmonic__, and this results in a decrease in the vibrational energy spacings with increasing vibrational energy.

In fact, the difference in energy between neighboring vibrational energy levels approaches zero as the vibrational energy approaches the dissociation energy. Figure 5.3.2 shows schematically a typical anharmonic potential energy surface along with its vibrational energy levels.

### Vibrational energies of an anharmonic oscillator

The vibrational energy levels of an anharmonic oscillator can be described by a power series expansion in \(\upsilon+\frac{1}{2}\)

\[ G_{\upsilon}=\omega_{e}\left ( \upsilon+\frac{1}{2} \right )-\omega_{e}\chi_{e}\left ( \upsilon+\frac{1}{2} \right )^2+ \; ... \]

where \(G_{\upsilon}\) is the energy of quantum level \(\upsilon\) in cm^{–1}, \(\omega_{e}\) is the "harmonic frequency" of the oscillator in cm^{–1}, and \(\chi_{e}\) is called the anharmonicity parameter. For the iodine states of interest in this experiment we can truncate the expansion at the second term, as indicated in equation 5.3.4. The difference between successive vibrational energies is then given by

\[ \Delta G_{\left ( \upsilon \right )} = \left| G_{\left ( \upsilon +1 \right )}-G_{\left ( \upsilon \right )}\right|=\omega_{e}-2\omega_{e}\chi_{e}\left ( \upsilon +1 \right ) \]

By measuring \(G_{\upsilon}\) for a given state, and plotting the change in vibrational energy spacing \(\Delta G_{\left ( \upsilon \right )}\) with respect to \(\upsilon +1\), both \(\omega_e\) and \(\omega_{e}\chi_{e}\) can be found. According to equation 5.3.5, this plot should be a straight line with a slope of \(-2\omega_{e}\chi_{e}\) and an intercept equal to \(\omega_e\). (Such a plot is called a Birge-Sponer plot. Hertha Sponer was a well-known and highly respected professor of physics at Duke University, one of only a few women physicists at major research universities during the first half of the twentieth century.)

### The Morse Potential

The Morse potential is a simple empirical potential with only two adjustable parameters that incorporates all of these features. In particular it gives rise to the two-term expression for \(G_{\upsilon}\) in equation 5.3.4, with the vibrational energies converging on the dissociation energy D_{e} (See Figure 5.3.2). The **Morse Potential **has the functional form

\[ V\left ( R \right )=D_{e}\left [ 1-e^{-a\left ( R-R_{e} \right )} \right ]^2 \]

where the Dissociation energy D_{e} and Morse parameter, a, are related to the parameters \(\omega_e\) and \(\omega_{e}\chi_{e}\) by

\[ a=\sqrt{\frac{8\pi^2 c\mu\omega_{e}\chi_{e}}{h}} \\ D_e =\frac{\omega_{e}}{4\chi_{e}} \]

Thus, for this potential, the constants, a, and D_{e} can be determined from a measurement of \(\omega_e\) and \(\omega_{e}\chi_{e}\). While other empirical potential functions provide a better representation of the true shape of the potential for many electronic states of diatomic molecules, for the electronic states of iodine considered here the Morse potential works quite well.

## Ground (X) and excited (B) states of iodine

In general, different electronic states have different potentials and different average interatomic separations. For some excited electronic states, the potential may not even have a minimum, or "bound" region. Excitation to one of these "repulsive" potentials would result directly in the dissociation of the molecule, and consequently the absorption spectrum would have no structure. In this experiment, you will focus on electronic transitions between the vibrational levels of two bound electronic states, the ground X \(^1\Sigma_{g}\) state and the excited \(B ^3\Pi_{u}\) state. Other electronic states of the iodine molecule will not be discussed here. The potential surfaces for the X and B states can be described by Morse functions. As the molecule dissociates along the ground state X surface, it will separate into two iodine atoms (I + I) in their ground \(^2P_{\tfrac{3}{2}}\) electronic states. However, in the electronically excited B state, dissociation leads to one iodine atom in its ground electronic \(^2P_{\tfrac{3}{2}}\) state, and the other in an excited \(^2P_{\tfrac{1}{2}}\) electronic state (I + I*) (see Figure 5.3.3 below). The energy for the electronic excitation of an iodine atom E(I*) is known quite accurately from atomic spectroscopy, the value being 7603 cm^{–1}. This energy is just the separation in energy between the iodine molecule X and B state potential curves in the limit where R approaches \(\infty\) (See Figure 5.3.3).

Electronic transitions caused by absorption or emission of light will occur between the various ground state vibrational levels \(\left ({\upsilon}^{\prime\prime}\right )\) and the excited state vibrational levels \(\left ({\upsilon}^{\prime}\right )\). Such transitions involving changes in both vibrational and electronic states are often called vibronic transitions. By convention the various parameters associated with the excited state are designated by \(^{\prime}\) and the parameters for the ground state are designated by \(^{\prime\prime}\), as shown in Figure 5.3.3. The frequency of a vibronic transition (in cm^{–1}) between a ground vibrational state level \(\left ({\upsilon}^{\prime\prime}\right )\) and an excited state vibrational level \(\left ({\upsilon}^{\prime}\right )\) will be given by

\[ \large \nu_{\left ( \upsilon^{\prime\prime},\upsilon^{\prime} \right )}=\nu_{el}+G_{\left ( \upsilon^{\prime} \right )}-G_{\left ( \upsilon^{\prime\prime} \right )} \]

where \(\nu_{el}\) is the difference in electronic energies of the two potentials at their respective minima, \(R_{e}^{\prime}\) and \(R_{e}^{\prime\prime}\) (see Figure 5.3.3). Note that the energy \(\nu_{el}\), the dissociation energies \(D_{e}^{\prime}\) and \(R_{e}^{\prime\prime}\), and the atomic excitation energy of the iodine atom E(I*) are related through the equation

\[ D_{e}^{\prime\prime}=\nu_{el}+D_{e}^{\prime}-E\left ( I^* \right ) \]

This can be seen in Figure 5.3.3.

In an absorption experiment, one usually identifies a series of bands (a "progression") that originate from a single ground state vibrational level \(\upsilon^{\prime\prime}\), say \(\upsilon^{\prime\prime}=0\), and end in a series of different excited state vibrational levels. These bands are then assigned to the appropriate \(\upsilon^{\prime}\) quantum numbers (not a trivial task!). If one then calculates the spacing between these bands, it can be seen from equation 5.3.8 that this will yield a linear equation just like equation 5.3.5, but where the independent variable is \(\left ( \upsilon^{\prime}+1\right )\). From the slope and intercept of this plot the values of \(\omega_{e}^{\prime}\) and \(\chi_{e}^{\prime}\) can be determined, and from these the values of the Morse parameters \(a^{\prime}\) and \(D_{e}^{\prime}\) for the excited state can be calculated using equations 5.3.7.

In an emission experiment, one usually pumps a single excited state \(\upsilon^{\prime}\), and then observes a band progression of transitions back to a number of different ground states \(\upsilon^{\prime\prime}\). These bands can then be assigned, and the separation between them plotted in the form of equation 5.3.5, with \(\left ( \upsilon^{\prime}+1\right )\) as the independent variable. This allows the ground state parameters \(\omega_{e}^{\prime\prime}\), \(\chi_{e}^{\prime\prime}\), \(a^{\prime\prime}\), and \(D_{e}^{\prime\prime}\) to be determined.

Consider the molecular potential-energy curves shown in Figure 5.3.3. They relate to different electronic states of the same molecule. Some of the vibrational energy levels of each electronic state are marked, together with some of the vibrational wavefunctions. Before absorption of light occurs, the molecule is most likely to be in the ground vibrational state of the lower electronic state, \(\upsilon^{\prime\prime}=0\), and so we shall concentrate on that initial level. The form of the lowest vibrational wavefunction shows that the most probable location of the nuclei is near their equilibrium separation \(R_{e}^{\prime\prime}\). However, at room temperature the \(\upsilon^{\prime\prime}=1\) and \(\upsilon^{\prime\prime}=2\) levels of iodine also have significant thermal populations and so electronic transitions originating from the \(\upsilon^{\prime\prime}=1\) and \(\upsilon^{\prime\prime}=2\) levels can also be seen; these are known as **hot bands**.

When an electronic transition occurs, the molecule is excited to the state represented by the upper curve in Figure 5.3.3. According to the Franck-Condon principle, the vibronic transitions occur so fast (on the order of femtoseconds or less) that the nuclear framework remains constant during the actual transition. We may, therefore, imagine the nuclear wavefunction of the molecule as rising up the vertical lines shown in Figure 5.3.3. This is the origin of the term *vertical transition*, which is used to denote an electronic transition that occurs without change of nuclear geometry.

The vertical transition cuts through several vibrational levels of the upper electronic state. The level marked \(\upsilon^{\prime}=25\) is the one in which the nuclei are most probably at the separation \(R_{e}^{\prime\prime}\) because the wavefunction has maximum amplitude there. Therefore this is the most probable level for the termination of the excitation. It is not, however the only vibrational level at which the transition may end, because several of its neighbors also have a high probability of having nuclei at the separation \(R_{e}^{\prime\prime}\). Therefore transitions take place to all vibrational levels in this region, but with greatest probability to the level with the wavefunction that overlaps the initial vibrational state wavefunction most favorably. Transitions from the \(\upsilon^{\prime\prime}=0\) state of iodine would tend to "miss" upper-level vibrational states with low values of \(\upsilon^{\prime}\). For this reason, these transitions are weak or not observed.

As stated above, the excited state vibrational wavefunctions which have the greatest Franck-Condon overlap with the \(\upsilon^{\prime\prime}=0\) ground state level, are those with the greatest amplitude near \(R_{e}^{\prime\prime}\). For these levels the maximum amplitude in the vibrational wavefunction occurs at the inner "classical turning point" or the edge of the potential well. Such wavefunctions lie at high values of \(\upsilon^{\prime}\) as shown in Figure 5.3.3. Thus we see that the vibrational structure of the spectrum depends on the relative displacement of the two molecular potential energy curves, and a long *progression* of vibrations (a lot of vibrational structure in the spectrum) is stimulated if the two electronic states are significantly displaced.

This Franck-Condon principle can be used to estimate the shift in the potential minimum on excitation, i.e. \(R_{e}^{\prime\prime}-R_{e}^{\prime}\). The strongest absorption transition (from \(\upsilon^{\prime\prime}=0\)) will be to some particular excited state level \(\upsilon^{*\prime}\). The separation, \(R^{*\prime}\), at the inner turning point of this exited state level can be identified. Assuming the shapes of both the ground and excited state potential curves are known, the two curves can be overlaid so that the minimum in the ground state \(R_{e}^{\prime\prime}\) overlaps vertically with the excited state turning point value \(R^{*\prime}\). The displacement, \(R_{e}^{\prime}-R_{e}^{\prime\prime}\), can then be read from the plots, or calculated analytically (see references 1 and 4).